A Finite Chiral 4-Polytope in $${\mathbb {R}}^4$$ R 4

Abstract

In this paper, we give an example of a chiral 4-polytope in projective 3-space. This example naturally yields a finite chiral 4-polytope in Euclidean 4-space, giving a counterexample to Theorem 11.2 of [8].

A chiral 4-polytope in R^3

“…The first known finite chiral 4-polytope in R 4 was discovered in [1] in 2014 proving false one half of the claim in [5]. In this paper we describe the first known infinite chiral 4-polytope in R 3 , proving false the remaining half of the claim.…”

Open problems on k-orbit polytopes

“…Regular polytopes of full rank were classified in [46,Theorem 11.2], where it is also stated that there are no chiral polytopes of full rank. This claim turned out to be false; there are both finite chiral 4-polytopes in E 4 [5] and infinite 4-polytopes in E 3 [66]. It is natural now to ask for the classification of full rank chiral polytopes.…”

“…Regular polytopes have been extensively studied; see [54] for the standard reference, and see [8,10,52,60] for a broad cross-section of current advances. Two-orbit polytopes (including chiral polytopes) have also received a lot of attention; see [36,72] for the basic notions and [5,11,17,34,43,65] for recent work. Very little is yet known about k-orbit polytopes for k ≥ 3.…”

On Roli's cube

“…Actually Roli's cube R isn't a cube, although it does share the 1-skeleton of a 4-cube. First described by Javier (Roli) Bracho, Isabel Hubard and Daniel Pellicer in [3], R is a chiral 4polytope of type {8, 3, 3}, faithfully realized in E 4 (a situation earlier thought impossible). Of course, Roli didn't himself name R; but the eponym is pleasing to his colleagues and has taken hold.…”